• The final type of design we will discuss today is the waitlist design, also known as the stepped-wedge design
• Their scientific value comes from their ability to track treatment effects among several subjects as they play out over time
• Waitlists play a diplomatic role because they overcome the problem of withholding treatment from a control group (remember our ethics discussion?)
• In this design, every subject is treated eventually; random assignment determines when they receive treatment
• It is a hybrid between a within-subject and a between-subject design, and it is widely used in public health interventions (e.g., vaccine rollouts)

Source: Hemming et al (2014)

• The immediate treatment effect is \(Y_{01} - Y_{00}\), that is, the effect of being treated in the current period but not in the preceding period
• We just need to take the numbers from the tables and apply inverse probability weighting again

\[ \begin{aligned} \widehat{E}[Y_{01} - Y_{00}] &= \frac{\frac{7 + 10}{0.25} + \frac{2 + 1}{0.25} + \frac{7 + 10}{0.25}}{\frac{2}{0.25} + \frac{2}{0.25} + \frac{2}{0.25}} \\ &- \frac{\frac{7 + 1 + 4 + 4 + 3 + 2 + 3}{0.75} + \frac{5 + 3 + 3 + 3}{0.50} + \frac{3 + 4}{0.25}}{\frac{6}{0.75} + \frac{4}{0.50} + \frac{2}{0.25}} = 2.72. \end{aligned} \]

• Finally, we will estimate the cumulative effect of TV advertising, which is \(Y_{11} - Y_{00}\), that is, the effect of being treated in both the preceding and current periods
• We do the same thing again, but now we consider the \(Y_{11}\) potential outcomes
• However, we must restrict our attention to the second and third weeks, because this type of treatment cannot occur in the first week

\[ \begin{aligned} \widehat{E}[Y_{11} - Y_{00}] &= \frac{\frac{9 + 8}{0.25} + \frac{4 + 10 + 10 + 3}{0.50}}{\frac{2}{0.25} + \frac{4}{0.50}} \\ &- \frac{\frac{5 + 3 + 3 + 3}{0.50} + \frac{3 + 4}{0.25}}{\frac{4}{0.50} + \frac{2}{0.25}} = 4.13. \end{aligned} \]

• Spillovers are a common problem in social science research
• They can be positive or negative, and they can be modelled explicitly in our analysis
• There are several methods to deal with spillovers, such as multi-level designs, within-subject designs, repeated-measures experiments, and waitlist designs
• Each design has its advantages and disadvantages, and the choice of design should be based on the research question and the context
• You can use DeclareDesign to simulate spillover designs and pretest posttest designs
• The statistical analysis of waitlist designs are a little tricky, but you can use the swCRTdesign package in R to help you with that